Tuesday, July 04, 2017

Voting Efficiency Gap: A Performative Theory?

Table 1: Distribution of Votes Among Parties and Districts

District

Tories

Whigs

Total

I

51

49

100

II

51

49

100

III

33

67

100

Total

135

165

300

1.0 Introduction

This post, amazingly enough, is on current events. Stephanopoulos and McGhee have developed a formula,
the efficiency gap, that measures the partisanship of the lines drawn for legislative districts.
In this post, I present a numerical illustration of this formula and connect it to current events.
I conclude with some questions.

2.0 Numerical Example

Consider a population of 300 voters divided between two parties. The Whigs are in the majority, with 55% of the electorate.
Suppose the government has a three-member council, with each member elected from a district. And
each district contains 100 voters.

2.1 Drawing Districts

The Tories, despite being the minority party have drawn the districts.
The votes in the last election are as in Table 1. The Tories are in the minority of the population, but
hold two out of three council seats.

The Tories, in this example, cannot win all seats. In the seats they lose, they want to pack as
many Whigs as possible. So where the Whigs win, they win overdominatingly. Many of the Whig votes
in that single district are wasted on running up a victory more than necessary. On the other hand,
the Tories try to draw their winning districts to win as narrowly as possible. The Whig
votes in the districts in which the Whigs lose are said to be cracked.

This is an extreme example, sensitive to small variations in the districts in which the Tories
win. They would probably want safer majorities in those districts.

As far as I can see, the drawing of odd-shaped district lines is not necessary for gerrymandering.
Consider a city surrounded by suburbs and a rural area. Suppose, that downtown tends to vote
differently than the suburbs and rural areas. One could imagine district lines drawn outward
from the central city. Depending on relative populations, that might distribute the urban
voters such that they predominate in all districts. On the other hand, one might create a few
compact districts in the center to pack many urban voters, with the ones remaining in cropped
pizza slices having their votes cracked.

2.2 Wasted Votes

Define a vote to be wasted if either it is for a losing candidate in your district or it is for a winning
candidate, but it exceeds the number needed for a majority in that district. The number of wasted
votes for each party in the numerical example is:

The Tories have 33 wasted votes.

The Whigs have 49 + 49 + (67 - 51) = 114 wasted votes.

The efficiency gap is a single number that combines the number of wasted votes
in both parties.
An invariance property arises here. As I have defined it, the number
of wasted votes, summed across parties, in each district is 49. Forty nine
is one less than half the number of votes in a district. This is no accident.

2.3 Arithmetic

In calculating the efficiency gap, one takes the absolute value of the difference
between the parties in the number of wasted votes. In the example, this
number is | 33 - 114 | = 81.

The efficiency gap is the ratio of this positive difference to the
number of voters. So the efficiency gap in the example
is 81/300 = 27%.

3.0 Contemporary Relevance in the United States

The United States Supreme Court has decided, in a number of cases over the last decades, that
gerrymandering might be something they can rule on. Partisan redistricting is not purely
a political issue that they do not want to get involved in. Apparently, however,
they have never found a clear example.

But what is gerrymandering? Can they define some sort of rule that lower courts can use?
How would politicians drawing up district lines know whether or not their decisions
will withstand challenges in court? Apparently, Justice Kennedy, among others expressed
a hankering for some such rule in his decision in
League of United Latin American Citizens (LULAC) vs. Perry (2006).

Gill vs. Whitford is
a current case on the Supreme Court docket. And the efficiency gap, which is relatively
new mathematics, may be discussed in the pleadings, at least, in this case.

So the creation of the mathematical formula illustrated above might affect the law
in the United States. If so, it will impact how districts are drawn and what
some consider fair. It is interesting that I can now raise the issue of
the performativity of mathematics in a non-historical context, while the
mathematics is, perhaps, performing.

4.0 Questions

I am working on reading two of the three references below. (Articles in law reviews
seem to be consistently lengthy.) I have some questions and comments.

Berstein and Duchin (2017) seems to raise some severe objections.
Suppose the election in a district with 100 voters is decided
either 75 to 25 or 76 to 24. The way I have defined it,
the difference in wasted votes in this district
is (24 - 25) or (25 - 24). That is, this district
contributes one vote to the difference in wasted votes.
So the definition of the efficiency gap privileges
races that are won with 75% of the vote.

Consider a case in which one party has support from 75 percent
of the voters. Suppose the districts are drawn such that each
district casts 75% of their votes for that party. So this
party wins 100% of the seats and the efficiency gap is
minimized. Do we want to say this is not an example of
gerrymandering?

Is the efficiency gap related
to powerindices somehow or other?
How should the efficiency gap be calculated if more than two parties are contesting an election? Mayhaps,
one should calculate the efficiency gap for each pair of parties. This loses the simplicity of a
single number. Also, sometimes clever Republican strategists might try to help themselves by helping the
Green Party, at the expense of the Democratic Party. How does this measure compare and contrast with
other measures? As I understand it, a measure of partisan swing, for example, relies on counterfactuals,
while the efficiency gap is not counterfactual.